Overview

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete. In the first part of the book the authors develop the relevant techniques, including the Weierstraß preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained. The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thuscan serve as source for special courses in singularity theory and local algebraic and analytic geometry.

Full Product Details

Author:   Gert-Martin Greuel ,  Christoph Lossen ,  Eugenii I. Shustin
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   Softcover reprint of hardcover 1st ed. 2007
Dimensions:   Width: 15.50cm , Height: 2.40cm , Length: 23.50cm
Weight:   0.741kg
ISBN:  

9783642066580

ISBN 10:   3642066585
Pages:   472
Publication Date:   12 February 2010
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
We will order this item for you from a manufatured on demand supplier.

Table of Contents

I. Singularity Theory. Basic Properties of Complex Spaces and Germs. Weierstrass Preparation and Finiteness Theorem. Application to Analytic Algebras. Complex Spaces. Complex Space Germs and Singularities. Finite Morphisms and Finite Coherence Theorem. Applications of the Finite Coherence Theorem. Finite Morphisms and Flatness. Flat Morphisms and Fibres. Singular Locus and Differential Forms. Hypersurface Singularities. Invariants of Hypersurface Singularities. Finite Determinacy. Algebraic Group Actions. Classification of Simple Singularities. Plane Curve Singularities. Parametrization. Intersection Multiplicity. Resolution of Plane Curve Singularities. Classical Topological and Analytic Invariants.- II. Local Deformation Theory. Deformations of Complex Space Germs. Deformations of Singularities. Embedded Deformations. Versal Deformations. Infinitesimal Deformations. Obstructions. Equisingular Deformations of Plane Curve Singularities.- Equisingular Deformations of the Equation. The Equisingularity Ideal. Deformations of the Parametrization. Computation of T^1 and T^2 . Equisingular Deformations of the Parametrization. Equinormalizable Deformations. Versal Equisingular Deformations.- Appendices: Sheaves. Commutative Algebra. Formal Deformation Theory. Literature.- Index.

Reviews

From the reviews: This monograph is dedicated to the theory of singularities, a subject with a central role in modern mathematics. ... This very well written book has a unified point of view based on the theory of analytic spaces, which allows a coherent presentation of both of its main themes: the theory of singularities and deformations of singularities. ... The book includes many examples and exercises ... . This monograph can serve as a source for several special courses in singularity theory and local analytic geometry. (Vasile Brinzanescu, Mathematical Reviews, Issue 2008 b)

From the reviews: This monograph is dedicated to the theory of singularities, a subject with a central role in modern mathematics. ! This very well written book has a unified point of view based on the theory of analytic spaces, which allows a coherent presentation of both of its main themes: the theory of singularities and deformations of singularities. ! The book includes many examples and exercises ! . This monograph can serve as a source for several special courses in singularity theory and local analytic geometry. (Vasile Brinzanescu, Mathematical Reviews, Issue 2008 b)

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